Generic simplicity of quantum Hamiltonian reductions

Abstract

Let a reductive group $G$ act on a smooth affine complex algebraic variety $X.$ Let $\mathfrak{g}$ be the Lie algebra of $G$ and $\mu :T^*(X)\rightarrow \mathfrak{g}^*$ be the moment map. If the moment map is flat, and for a generic character $\chi :\mathfrak{g}\rightarrow \mathbb{C}$, the action of $G$ on $\mu ^{-1}(\chi )$ is free, then we show that for very generic characters $\chi $ the corresponding quantum Hamiltonian reduction of the ring of differential operators $D(X)$ is simple.